A chemical-isotopic equilibrium model was developed for adjustment of radiocarbon ages of groundwater from the arid Tucson basin for dilution of the initial groundwater C-14 activity by the solution of soil calcite having a C-14 of 25 ± 19% modern. Input to the model consisted of the laboratory chemical analyses for Ca⁺⁺, Mg⁺⁺, Na⁺, H₄SiO₄, SO₄⁼, HCO₃⁻, CO₃⁼, NO₃⁻, and pH, and δ C-13 for the total dissolved carbon in the groundwater. Output consisted of the equilibrium chemical composition of the groundwater, the ratio of soil CO₂ derived to total dissolved carbon, Q, and δ C-13 of total dissolved carbon, H₂CO₃, HCO₃⁻, and CO₃⁼, and δ C-13 for the soil CO₂ and calcite that initially dissolved in the surface water as it equilibrated with soil minerals. Radiocarbon age of the groundwater is computed from the equation T = 8270 ln [(Q + (1-Q) A(CaCO₃)/Am] where T is the age in years before A.D. 1950, A(CaCO₃) is the soil calcite activity and Am is the measured activity for the dissolved carbonate in the groundwater, both with respect to modern wood. The validity of the model was tested by comparing the predicted values for δ C-13 (CO₂), δ C-13 (CaCO₃) with measured values for samples from the Tucson basin. δ C-13 (CO 2) calculated = (-12.9 ± 1.9) per mil PDB. δ C-13 (CO2) measured = (-15.1 ± 2.8) per mil PDB. δ C-13 (CaCO3) calculated = (-3.9 ± 1.7) per nil PDB. δ C-13 (CaCO3) measured = (-3.6 ± 1.7) per mil PDB. On the basis of these results, the model adequately describes the natural system and may prove useful in future radiocarbon dating work in desert regions.

Phase I Draft. Prepared for United States Army Corps of Engineers, Los Angeles District, Tucson Urban Study, Regional Flood Control Element, by The University of Arizona, College of Earth Sciences, Water Resources Research Center in cooperation with College of Agriculture.

Two digital- computer models of the ground -water reservoir of the Tucson basin, in south - central Arizona, were constructed to study errors in digital models and to evaluate the worth of additional basic data to models. The two models differ primarily in degree of detail -- the large -scale model consists of 1,890 nodes, at a 1/2 -mile spacing; and the small -scale model consists of 509 nodes, at a l -mile spacing. Potential errors in the Tucson basin models were classified as errors associated with computation, errors associated with mathematical assumptions, and errors in basic data: the model parameters of coefficient of storage and transmissivity, initial water levels, and discharge and recharge. The study focused on evaluating the worth of additional basic data to the small -scale model. A basic form of statistical decision theory was used to compute expected error in predicted water levels and expected worth of sample data (expected reduction in error) over the whole model associated with uncertainty in a model variable at one given node. Discrete frequency distributions with largely subjectively- determined parameters were used to characterize tested variables. Ninety -one variables at sixty - one different locations in the model were tested, using six separate error criteria. Of the tested variables, 67 were chosen because their expected errors were likely to be large and, for the purpose of comparison, 24 were chosen because their expected errors were not likely to be particularly large. Of the uncertain variables, discharge /recharge and transmissivity have the largest expected errors (averaging 155 and 115 feet, respectively, per 509 nodes for the criterion of absolute value of error) and expected sample worths (averaging 29 and 14 feet, respectively, per 509 nodes). In contrast, initial water level and storage coefficient have lesser values. Of the more certain variables, transmissivity and initial water level generally have the largest expected errors (a maximum of 73 per feet per 509 nodes) and expected sample worths (a maximum of 12 feet per 509 nodes); whereas storage coefficient and discharge/ recharge have smaller values. These results likely are not typical of those from many ground -water basins, and may apply only to the Tucson basin. The largest expected errors are associated with nodes at which values of discharge /recharge are large or at which prior estimates of transmissivity are very uncertain. Large expected sample worths are associated with variables which have large expected errors or which could be sampled with relatively little uncertainty. Results are similar for all six of the error criteria used. Tests were made of the sensitivity of the method to such simplifications and assumptions as the type of distribution function assumed for a variable, the values of the estimated standard deviations of the distributions, and the number and spacing of the elements of each distribution. The results are sensitive to all of the assumptions and therefore likely are correct only in order of magnitude. However, the ranking of the types of variables in terms of magnitude of expected error and expected sample worth is not sensitive to the assumptions, and thus the general conclusions on relative effects of errors in different variables likely are valid. Limited studies of error propagation indicated that errors in predicted water levels associated with extreme erroneous values of a variable commonly are less than 4 feet per node at a distance of 1 mile from the tested node. This suggests that in many cases, prediction errors associated with errors in basic data are not a major problem in digital modeling.

No description available.

wo digital-computer models of the ground-water reservoir of the Tucson basin, in south-central Arizona, were constructed to study errors in digital models and to evaluate the worth of additional basic data to models. The two models differ primarily in degree of detail -- the large-scale model consists of 1,890 nodes, at a 1/2-mile spacing; and the small-scale model consists of 509 nodes, at a 1-mile spacing. Potential errors in the Tucson basin models were classified as errors associated with computation, errors associated with mathematical assumptions, and errors in basic data: the model parameters of coefficient of storage and transmissivity, initial water levels, and discharge and recharge. The study focused on evaluating the worth of additional basic data to the small-scale model. A, basic form of statistical decision theory was used to compute expected error in predicted water levels and expected worth of sample data (expected reduction in error) over the whole model associated with uncertainty in a model variable at one given node. Discrete frequency distributions with largely subjectively-determined parameters were used to characterize tested variables. Ninety-one variables at sixtyone different locations in the model were tested, using six separate error criteria. Of the tested variables, 67 were chosen because their expected errors were likely to be large and, for the purpose of comparison, 24 were Chosen because their expected errors were not likely to be particularly large. Of the uncertain variables, discharge/recharge and transmissivity have the largest expected errors (averaging 155 and 115 feet, respectively, per 509 nodes for the criterion of absolute value of error) and expected sample worths (averaging 29 and 14 feet, respectively, per 509 nodes). In contrast, initial water level and storage coefficient have lesser values. Of the more certain variables, transmissivity and initial water level generally have the largest expected errors (a maximum of 73 per feet per 509 nodes) and expected sample worths (a maximum of 12 feet per 509 nodes); whereas storage coefficient and discharge/ recharge have smaller values. These results likely are not typical of those from many ground-water basins, and may apply only to the Tucson basin. The largest expected errors are associated with nodes at which values of discharge/recharge are large or at which prior estimates of transudssivity are very uncertain. Large expected sample worths are associated with variables which have large expected errors or which could be sampled with relatively little uncertainty. Results are similar for all six of the error criteria used. Tests were made of the sensitivity of the method to such simplifications and assumptions as the type of distribution function assumed for a variable, the values of the estimated standard deviations of the distributions, and the number and spacing of the elements of each distribution. The results are sensitive to all of the assumptions and therefore likely are correct only in order of magnitude. However, the ranking of the types of variables in terms of magnitude of expected error and expected sample worth is not sensitive to the assumptions, and thus the general conclusions on relative effects of errors in different variables likely are valid. Limited studies of error propagation indicated that errors in predicted water levels associated with extreme erroneous values of a variable commonly are less than 4 feet per node at a distance of 1 mile from the tested node. This suggests that in many cases, prediction errors associated with errors in basic data are not a major problem in digital modeling.

Completion Report / Contract No. 26-235-816-40-2-050-0735 / Arizona Department of Health Services / Bureau of Water Quality Control

No description available.

The Tucson Basin is a semi-arid alluvial basin in southeastern Arizona in which the Santa Cruz River, an ephemeral stream, flows south to north with its flows resulting directly from rainfall. The City of Tucson discharges treated sewage effluent into the bed of the Santa Cruz and to some irrigated farms. Previous investigations indicate that sewage effluent is recharging the Tucson Basin Aquifer with the water spreading horizontally in the Fort Lowell Formation. The ¹⁸0/¹⁶0 ratios determined in water samples by the author support the findings of these previous investigations. Sewage effluent had an average δc0-18 value of -7.9 per mil and water samples from the north Santa Cruz wells had an average δc0-18 value of -9.3 per mil. Up hydraulic gradient, the ¹⁸0/¹⁶0 ratios are lighter indicating that sewage recharge water has mixed with ground water. In the case of one well in the mixed zone, it is calculated that approximately 70 percent of the water comes from sewage recharge and 30 percent from normal ground water. Recharge water spreads horizontally in the Fort Lowell Formation up to two miles on each side of the river. The δc0-18 values of water samples from the south Santa Cruz wells averaged -8.9 per mil and compared closely to the average δc0-18 values for summer flows in the Santa Cruz River of -8.2 per mil.

The concept of hydrochemical facies is used to study the distribution and, indirectly, to identify the origin of the chemical character of ground water in the basin-fill aquifer of the Tucson Basin in relationship to the hydrogeologic framework. Hydrochemical fades of ground water is defined operationally in terms of the lateral (horizontal) variation of chemical quality. The following chemical constituents are included in this study: total dissolved solids, calcium, magnesium, sodium, potassium, chloride, sulfate, bicarbonate, pH, nitrate, fluoride, strontium and zinc. A conceptual process-response model relates the regional distribution of dissolved constituents to the following hydrogeologic controls: (a) the chemical composition of the rock and soil in the drainage area before recharge to the ground-water basin and conditions at the recharge sites, (b) the lithology of the basin-fill aquifer, and (c) the direction of groundwater flow within the aquifer itself. Trend surface analysis suggests that the regional distribution patterns of total dissolved solids, calcium, sodium, sulfate and strontium show a tendency to parallel the direction of ground-water flow. The distribution pattern of chloride ions based on old analyses shows a trend opposite to the distribution pattern of chloride ions based on new analyses from the same area. Nitrate content of ground water and specific capacity of wells seem to be related. Q-factor analysis of data from the basin-fill aquifer demonstrates that the overall chemical character of the ground water does not change substantially as it moves through the basin. It is concluded that the chemical character of ground water in the basin-fill aquifer of the Tucson Basin was acquired mainly during contact with various rock types in the drainage basin before recharge. The lithology of the aquifer, presumably, only plays a secondary role in determining the overall chemical composition of the ground water. After recharge to the basin-fill aquifer the distribution of dissolved constituents is controlled primarily by the flow pattern. A chemical equilibrium model of calcite and water is used to approach the problem of determining whether precipitation or dissolution of calcite takes place in the aquifer. Measured calcium ion concentrations and pH values are compared to calcium ion concentrations and pH values computed for the equilibrium model. Assuming that the equilibrium model represents actual conditions in the aquifer, departures from the equilibrium model may be used to predict the chemical behaviour of calcite In the basin-fill aquifer.

This dissertation is concerned with the development of mathematical aquifer models that combine hydrological, hydrochemical and isotopic data. One prerequisite for the construction of such models is that prior information about the variables and parameters be quantified in space and time by appropriate statistical methods. Various techniques using multivariate statistical data analyses and geostatistical methods are examined in this context. The available geostatistical methods are extended to deal with the problem at hand. In particular, a three-dimensional interactive geostatistical package has been developed for the estimation of intrinsic and nonintrinsic variables. This package is especially designed for groundwater applications and incorporates a maximum likelihood cross-validation method for estimating the parameters of the covariance function. Unique features of this maximum likelihood cross-validation method include: the use of an adjoint state method to compute the gradient of the likelihood function, the computation of the covariance of the parameter estimates and the use of identification criteria for the selection of a covariance model. In addition, it can be applied to data containing measurement errors, data regularized over variable lengths, and to nonintrinsic variables. The above methods of analysis are applied to synthetic data as well as hydrochemical and isotopic data from the Tucson aquifer in Arizona and the Madrid Basin in Spain. The dissertation also includes a discussion of the processes affecting the transport of dissolved constituents in groundwater, the mathematical formulation of the inverse solute transport problem and a proposed numerical method for its solution.